Wave propagation methods for hyperbolic problems on mapped
grids
A France-Taiwan Orchid Project Progress Report 2008-2009
Keh-Ming Shyue
Department of Mathematics National Taiwan University
Taiwan
Talk objective
Review a body-fitted mapped grid approach for
numerical approximation of hyperbolic balance laws in multi-D with complex geometries
Present numerical results for problems arising from Compressible inviscid gas flow
Barotropic cavitating flow
Shallow granular avalanches
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 2/47
Mathematical model
Consider a hyperbolic balance laws of the form
∂
∂t q (~x, t) +
N
X
j=1
∂
∂x j f j (q, ~x) = ψ(q)
in a general multidimensional domain
~x = (x 1 , x 2 , . . . , x N ) : spatial vector, t : time
q ∈ lR m : vector of m state quantities
f j ∈ lR m : flux vector, ψ ∈ lR m : source terms
Model is assumed to be hyperbolic, where P N j=1 α j (∂f j /∂q) is
diagonalizable with real e-values ∀α j ∈ lR
Mathematical model (Cont.)
In a body-fitted mapped grid approach, we introduce a coordinate change ~x 7→ ~ ξ via
ξ = (ξ ~ 1 , ξ 2 , . . . , ξ N ) , ξ j = ξ j (~x)
that transform a physical domain Ω to a logical one Ω ˆ , see below when N = 2 ,
−1 0 1
−1.5
−1
−0.5 0
−1 −0.5 0 0.5
−2
−1.5
−1
−0.5
x 1 x 2
ξ 1
ξ 2
Ω −→ Ω ˆ
mapping
ξ 1 = ξ 1 (x 1 , x 2 ) ξ 2 = ξ 2 (x 1 , x 2 )
logical domain physical domain
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 4/47
Mathematical model (Cont.)
In a body-fitted mapped grid approach, we introduce a coordinate change ~x 7→ ~ ξ via
ξ = (ξ ~ 1 , ξ 2 , . . . , ξ N ) , ξ j = ξ j (~x)
that transform a physical domain Ω to a logical one Ω ˆ , and so equations into the form
∂q
∂t +
N
X
j=1
∂ ˜ f j
∂ξ j = ψ(q)
with
f ˜ j =
N
X
k=1
f k ∂ξ j
∂x k
Mathematical model (Cont.)
Basic coordinate mapping relations in N = 3 are
1 0 0 0
0 ∂ x 1 ξ 1 ∂ x 2 ξ 1 ∂ x 3 ξ 1 0 ∂ x 1 ξ 2 ∂ x 2 ξ 2 ∂ x 3 ξ 2 0 ∂ x 1 ξ 3 ∂ x 2 ξ 3 ∂ x 3 ξ 3
= 1 J
J 0 0 0
0 J 11 J 21 J 31 0 J 12 J 22 J 32 0 J 13 J 23 J 33
where J = |∂(x 1 , x 2 , x 3 )/∂(ξ 1 , ξ 2 , ξ 3 )| = det (∂(x 1 , x 2 , x 3 )/∂(ξ 1 , ξ 2 , ξ 3 )), J 11 =
∂(x 2 , x 3 )
∂(ξ 2 , ξ 3 )
, J 21 =
∂(x 1 , x 3 )
∂(ξ 3 , ξ 2 )
, J 31 =
∂(x 1 , x 2 )
∂(ξ 2 , ξ 3 )
, J 12 =
∂(x 2 , x 3 )
∂(ξ 3 , ξ 1 )
, J 22 =
∂(x 1 , x 3 )
∂(ξ 1 , ξ 3 )
, J 32 =
∂(x 1 , x 2 )
∂(ξ 3 , ξ 1 )
, J 13 =
∂(x 2 , x 3 )
∂(ξ 1 , ξ 2 )
, J 23 =
∂(x 1 , x 3 )
∂(ξ 2 , ξ 1 )
, J 33 =
∂(x 1 , x 2 )
∂(ξ 1 , ξ 2 )
.
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 5/47
Compressible Euler equations
Cartesian coordinate case
∂
∂t
ρ ρu i
E
+
N
X
j=1
∂
∂x j
ρu j
ρu i u j + pδ ij Eu j + pu j
=
0
−ρ∂ x i φ
−ρ~u · ∇φ
, i = 1, . . . , N
Generalized coordinate case
∂
∂t
ρ ρu i
E
+
N
X
j=1
∂
∂ξ j
ρU j
ρu i U j + p∂ x i ξ j EU j + pU j
=
0
−ρ∂ x i φ
−ρ~u · ∇φ
ρ : density, p = p(ρ, e) : pressure , e : internal energy E = ρe + ρ P N
j=1 u 2 j /2 : total energy, φ : gravitational potential
U j = ∂ t ξ j + P N
i=1 u i ∂ x i ξ j : contravariant velocity in ξ j -direction
Finite volume approximation
Employ finite volume formulation of numerical solution
Q n ijk ≈ 1
∆ξ 1 ∆ξ 2 ∆ξ 3 Z
C ijk
q(ξ 1 , ξ 2 , ξ 3 , t n ) dV
that gives approximate value of cell average of solution q over cell C ijk at time t n (sample case in 2D shown below)
i − 1 i − 1
i
i j
j
j + 1 j + 1
C ij C ˆ ij
ξ 1 ξ 2
mapping
∆ξ 1
∆ξ 2 logical domain physical domain
←−
x 1 = x 1 (ξ 1 , ξ 2 ) x 2 = x 2 (ξ 1 , ξ 2 ) x 1
x 2
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 7/47
Finite volume (Cont.)
In three dimensions N = 3 , equations to be solved take
∂q
∂t +
N
X
j=1
∂ ˜ f j
∂ξ j = ψ(q)
A simple dimensional-splitting method based on wave propagation approach of LeVeque et al. is used, i.e. ,
Solve one-dimensional Riemann problem normal at each cell interfaces
Use resulting jumps of fluxes (decomposed into each wave family) of Riemann solution to update cell
averages
Introduce limited jumps of fluxes to achieve high
resolution
Finite volume (Cont.)
Basic steps of a dimensional-splitting scheme ξ 1 -sweeps: solve
∂q
∂t + ∂ ˜ f 1
∂ξ 1 = 0 updating Q n ijk to Q ∗ ijk
ξ 2 -sweeps: solve
∂q
∂t + ∂ ˜ f 2
∂ξ 2 = 0 updating Q ∗ ijk to Q ∗∗ ijk
ξ 3 -sweeps: solve
∂q
∂t + ∂ ˜ f 3
∂ξ 3 = 0, updating Q ∗∗ ijk to Q n+1 ijk
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 9/47
Finite volume (Cont.)
Consider ξ 1 -sweeps, for example, First order update is
Q ∗ ijk = Q n ijk − ∆t
∆ξ 1
h A + 1 ∆Q n
i−1/2,jk + A − 1 ∆Q n
i+1/2,jk
i
with the fluctuations
(A + 1 ∆Q) n i−1/2,jk = X
m:(λ 1,m ) n i−1/2,jk >0
(Z 1,m ) n i−1/2,jk
and
(A − 1 ∆Q) n i+1/2,jk = X
m:(λ 1,m ) n i+1/2,jk <0
(Z 1,m ) n i+1/2,jk
(λ 1,m ) n ι−1/2,jk & (Z 1,m ) n ι−1/2,jk are in turn wave speed and f -waves
for the mth family of the 1D Riemann problem solutions
Finite volume (Cont.)
High resolution correction is
Q ∗ ijk := Q ∗ ijk − ∆t
∆ξ 1
˜ F 1 n
i+1/2,jk − ˜ F 1 n
i−1/2,jk
with ( ˜ F 1 ) n i−1/2,jk = 1 2
m w
X
m=1
sign (λ 1,m )
1 − ∆t
∆ξ 1 |λ 1,m |
Z ˜ 1,m
n
i−1/2,jk
Z ˜ ι,m is a limited value of Z ι,m
It is clear that this method belongs to a class of upwind schemes, and is stable when the typical CFL (Courant-Friedrichs-Lewy) condition:
ν = ∆t max m (λ 1,m , λ 2,m , λ 3,m )
min (∆ξ 1 , ∆ξ 2 , ∆ξ 3 ) ≤ 1,
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 11/47
Smooth vortex flow: accuracy test
Compressible Euler equations with ideal gas law Initial condition for vortex is set by
ρ =
1 − 25(γ − 1)
8γπ 2 exp (1 − r 2 )
1/(γ−1)
, p = ρ γ ,
u 1 = 1 − 5
2π exp ((1 − r 2 )/2) (x 2 − 5) , u 2 = 1 + 5
2π exp ((1 − r 2 )/2) (x 1 − 5) , r = p(x 1 − 5) 2 + (x 2 − 5) 2
kE z k 1 ,∞ = kz comput − z exact k 1 ,∞ : discrete 1 - or
maximum-norm for state variable z
Smooth vortex flow: accuracy test
Density contours on a 40 × 40 grid at time t = 10
0 2 4 6 8 10
0 2 4 6 8 10
0 2 4 6 8 10
0 2 4 6 8
Cartesian grid 10 Quadrilateral grid
x 1
x 1
x 2
x 2
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 13/47
Smooth vortex flow: accuracy test
Cartesian grid results
Mesh size 40 × 40 80 × 80 160 × 160 320 × 320 Order
kE ρ k 1 7.0710(−1) 1.9186(−1) 4.7927(−2) 1.1941(−2) 1.97
kE p k 1 8.5264(−1) 2.3594(−1) 5.9209(−2) 1.4721(−2) 1.96
kE u 1 k 1 2.3716(00) 6.1437(−1) 1.5298(−1) 3.8204(−2) 1.99
kE u 2 k 1 1.9377(00) 4.7673(−1) 1.1773(−1) 2.9262(−2) 2.02
kE ρ k ∞ 1.4587(−1) 3.8860(−2) 9.5936(−3) 2.3179(−3) 2.00
kE p k ∞ 1.8528(−1) 5.0122(−2) 1.2401(−2) 3.0285(−3) 1.98
kE u 1 k ∞ 3.9934(−1) 1.0488(−1) 2.4857(−2) 6.1654(−3) 2.01
kE u 2 k ∞ 2.0948(−1) 5.5860(−2) 1.3506(−2) 3.3386(−3) 2.00
Smooth vortex flow: accuracy test
Quadrilateral grid results
Mesh size 40 × 40 80 × 80 160 × 160 320 × 320 Order kE ρ k 1 1.1921(00) 4.1732(−1) 1.6058(−1) 7.0078(−2) 1.36 kE p k 1 1.4984(00) 5.3128(−1) 2.1063(−1) 9.3740(−2) 1.33 kE u 1 k 1 2.7085(00) 8.5937(−1) 2.8118(−1) 1.0743(−1) 1.56 kE u 2 k 1 2.3014(00) 7.4990(−1) 2.7248(−1) 1.1608(−1) 1.44 kE ρ k ∞ 1.8793(−1) 6.2063(−2) 1.9104(−2) 7.0730(−3) 1.59 kE p k ∞ 2.2841(−1) 7.2502(−2) 2.1285(−2) 7.9266(−3) 1.63 kE u 1 k ∞ 4.0762(−1) 1.3207(−1) 4.2383(−2) 1.5737(−2) 1.57 kE u 2 k ∞ 2.6456(−1) 9.0362(−2) 2.7416(−2) 1.2385(−2) 1.50
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 15/47
Shock waves over circular array
A Mach 1.42 shock wave in water over a circular array
t=0
shock
Shock waves over circular array
Grid system
−3 −2 −1 0 1 2 3 4 5
−4
−3
−2
−1 0 1 2 3 4
time=0
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 17/47
Shock waves over circular array
Contours for density
t=0.0024
Steady state shock tracking
A Mach 3 compressible gas flow over a 20 o ramp
Exact shock location (an oblique shock with 37.8 o ) is inserted into underlying grid for computation
0.0 0.2 0.4 0.6 0.8 1.0
2.02.22.42.62.83.0
Grid
M a c h
x
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 19/47
Compressible Multiphase Flow
Homogeneous equilibrium pressure & velocity across material interfaces
Volume-fraction based model equations (Shyue JCP
’98, Allaire et al. JCP ’02)
∂
∂t (α i ρ i ) + 1 J
N d
X
j=1
∂
∂ξ j (α i ρ i U j ) = 0, i = 1, 2, . . . , m f
∂
∂t (ρu i ) + 1 J
N d
X
j=1
∂
∂ξ j (ρu i U j + pJ ji ) = 0, i = 1, 2, . . . , N d ,
∂E
∂t + 1 J
N d
X
j=1
∂
∂ξ j (EU j + pU j ) = 0,
∂α i
∂t + 1 J
N d
X
j=1
U j ∂α i
∂ξ j = 0, i = 1, 2, . . . , m f − 1;
Moving cylindrical vessel
Initial setup
time=0
air helium
interface
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 21/47
Moving cylindrical vessel
Solution at time t = 0.25
Moving cylindrical vessel
Solution at time t = 0.5
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 23/47
Moving cylindrical vessel
Solution at time t = 0.75
Moving cylindrical vessel
Solution at time t = 1
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 25/47
Moving cylindrical vessel
cross-sectional plot along boundary
0 1 2
0 1 2 3
ρ
0 1 2
0 1 2 3
0 1 2
0 0.5 1 1.5
0 1 2
0 0.5 1 1.5
0 1 2
0 1 2 3 4
p
0 1 2
0 1 2 3 4
0 1 2
0.5 1 1.5
0 1 2
0 0.5 1 1.5 2
t = 0.25 t = 0.5 t = 0.75 t = 1
θ(π) θ(π)
θ(π)
θ(π)
Barotropic cavitating flow
A relaxation model of Saurel et al. (JCP ’090
∂
∂t (α 1 ρ 1 ) +
N
X
j=1
∂
∂x j (α 1 ρ 1 u j ) = 0,
∂
∂t (α 2 ρ 2 ) +
N
X
j=1
∂
∂x j (α 2 ρ 2 u j ) = 0,
∂
∂t (ρu i ) +
N
X
j=1
∂
∂x j (ρu i u j + pδ ij ) = 0, i = 1, . . . , N
∂α 1
∂t +
N
X
j=1
u j ∂α 1
∂x j = 1
µ (p 1 (ρ 1 ) − p 2 (ρ 2 ))
Each phasic pressure p ι satisfies p ι (ρ) = (p 0 + B) (ρ/ρ 0 ) γ − B
Mixture pressure p satisfies p = α 1 p 1 + α 2 p 2 , µparameter
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 27/47
Mixture speed of sound
Wood formula (stiffness in c ¯ vs. α )
1
ρ¯ c 2 = α ρ w c 2 w
+ 1 − α ρ g c 2 g
ρ w = 10 3 kg/m 3 , c w = 1449.4m/s, ρ g = 1.0kg/m 3 , c g = 374.2m/s
0 0.2 0.4 0.6 0.8 1
0 500 1000 1500
¯c
Cavitation test: 1D
Homogeneous gas-liquid mixture with α g = 10 −2
u 1 = −100 m/s u 1 = 100 m/s
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 29/47
Cavitation test: 1D
Formation of cavitating gas bubble
−1 0 0 1
5
10 x 10
4−1 0 1
−100
−50 0 50 100
u 1
−1 0 0 1
500 1000
ρ
−1 0 0 1
0.5 1
cavitation
p α
x
x
Cavitation test: 2D steady state
Uniform gas-liquid mixture with speed 600 m /s over a circular region
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 31/47
Cavitation test: 2D steady state
Pseudo colors of volume fraction
Formation of cavitation zone (Onset–shock induced,
diffusion, . . . ?)
Cavitation test: 2D steady state
Pseudo colors of pressure
Smooth transition across liquid-gas phase boundary
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 33/47
Cavitation test: 2D steady state
Pseudo colors of volume fraction: 2 circular case
Convergence of solution as the mesh is refined ?
t=0.05
Shallow granular avalanches
Depth-average Savage-Hutter equations
∂h
∂t +
N
X
j=1
∂
∂x j (hu j ) = 0,
∂
∂t (hu i ) +
N
X
j=1
∂
∂x j
hu i u j + 1
2 β x h 2 δ ij
= hψ i , i = 1, . . . , N,
where Mohr-Coulomb closure is used with β x = K x cos ζ,
K x =
K x − if ∇ · ~u > 0
K x + if ∇ · ~u < 0, K x ± = 2
cos 2 φ 1 ± r
1 − cos 2 φ cos 2 δ
!
− 1,
ψ i = sin ζδ 1i − u i
|~u| tan δ cos ζ + κu 2 1 − cos ζ ∂B
∂x i
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 35/47
Earth pressure coefficients
Jump discontinuity on K x , |K x + − K x − | 6= 0 (see below where δ = 30 o is used as a reference)
30 35 40 45 50 55 60
0 2 4 6 8 10 12 14
K x +
K x
−
φ (degree)
K ± x
Avalanche on an inclined channel
Hemispherical granular material
Parameters: ζ = 35 o , φ = 30 o , δ = 30 o t = 0
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 37/47
Avalanche on an inclined channel
Contour plots for granular height (normal to channel)
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 0
Avalanche on an inclined channel
Down-flow phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 3
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 38/47
Avalanche on an inclined channel
Down-flow phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 6
Avalanche on an inclined channel
Down-flow phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 9
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 38/47
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 12
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 15
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 38/47
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 18
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 21
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 38/47
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 24
Avalanche on an inclined channel
Cross-sectional plot along the channel
t = 0
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 39/47
Avalanche on an inclined channel
Down-flow phase
t = 3
Avalanche on an inclined channel
Down-flow phase
t = 6
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 39/47
Avalanche on an inclined channel
Down-flow phase
t = 9
Avalanche on an inclined channel
Deposit phase
t = 12
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 39/47
Avalanche on an inclined channel
Deposit phase
t = 15
Avalanche on an inclined channel
Deposit phase
t = 18
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 39/47
Avalanche on an inclined channel
Deposit phase
t = 21
Avalanche on an inclined channel
Deposit phase
t = 24
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 39/47
Avalanche on an inclined channel
Hemispherical granular material
Parameters: ζ = 35 o , φ = 40 o , δ = 30 o
t = 0
Avalanche on an inclined channel
Contour plots for granular height (normal to channel)
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 0
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 41/47
Avalanche on an inclined channel
Down-flow phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 3
Avalanche on an inclined channel
Down-flow phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 6
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 41/47
Avalanche on an inclined channel
Down-flow phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 9
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 12
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 41/47
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 15
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 18
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 41/47
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 21
Avalanche on an inclined channel
Deposit phase
0 5 10 15 20 25 30
−6
−4
−2 0 2 4 6
t = 24
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 41/47
Avalanche on an inclined channel
Cross-sectional plot along the channel
t = 0
Avalanche on an inclined channel
Down-flow phase
t = 3
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 42/47
Avalanche on an inclined channel
Down-flow phase
t = 6
Avalanche on an inclined channel
Down-flow phase
t = 9
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 42/47
Avalanche on an inclined channel
Deposit phase
t = 12
Avalanche on an inclined channel
Deposit phase
t = 15
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 42/47
Avalanche on an inclined channel
Deposit phase
t = 18
Avalanche on an inclined channel
Deposit phase
t = 21
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 42/47
Avalanche on an inclined channel
Deposit phase
t = 24
Avalanche on an inclined channel
Pseudo colors of velocity divergence: Down-flow phase
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 43/47
Avalanche on an inclined channel
Pseudo colors of velocity divergence: Down-flow phase
Avalanche on an inclined channel
Pseudo colors of velocity divergence: Deposit phase
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 43/47
Avalanche on an inclined channel
Pseudo colors of velocity divergence: Deposit phase
Steady state ramp computation
A Froude 7 shallow granular flow over a 24.9 o ramp Parameters: ζ = 32.6 o , φ = 38 o , δ = 31 o
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 44/47
Steady state ramp computation
A Froude 7 shallow granular flow over a 24.9 o ramp
Parameters: ζ = 32.6 o , φ = 38 o , δ = 31 o
Steady state ramp computation
Cross-sectional plot along ramp with three different φ
0 0.5 1 1.5 2
0
5 φ=31
φ =38 φ =50
0 0.5 1 1.5 2
0 20 40
0 0.5 1 1.5 2
−500 0 500
x
h p ∇ ·~u
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 46/47
Future direction
Numerical methodology
Vacuum (dry) state treatment
Flux & source terms well-balanced
Interface sharping by techniques such as Lagrange-like moving mesh or front tracking
· · ·
Applications
Relaxation model as applied to more practical cavitation problems
General depth-average models to granular flows
· · ·
Future direction
Numerical methodology
Vacuum (dry) state treatment
Flux & source terms well-balanced
Interface sharping by techniques such as Lagrange-like moving mesh or front tracking
· · ·
Applications
Relaxation model as applied to more practical cavitation problems
General depth-average models to granular flows
· · ·
Thank You
ISCM II & EPMESC XII, 29 November–3 December, 2009 – p. 47/47
